For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows: $$ \phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - P(G)| : F \in \mathscr{F} , G \in \mathscr{G}, P(F)>0 \} $$

Now my question is how does one go about conditioning $\phi(\mathscr{F},\mathscr{G})$ with respect to another $\sigma-$field $\mathscr{H}$, that is, $\phi(\mathscr{F},\mathscr{G}| \mathscr{H})$.

Any help will be appreciated.